Distances in sets of positive Kor\'anyi upper density in Heisenberg Group

Abstract

We prove that any measurable set in the Heisenberg group, $\mathbb{H}^n$, of positive upper density has the property that all sufficiently large real numbers are realised as the Kor\'anyi distance between points in that set. The result can be seen as a Heisenberg group analogue to a corresponding Euclidean large distance set result in the $1986$ paper of Bourgain, \cite{1986Bourgain}. Along the way, to prove our main theorem, we give the ``decay" of the coefficients $R_{k}(\lambda, \sigma)$, appearing in the spectral decomposition of the group Fourier transform, $\hat{\sigma}(\lambda)$ $= \sum_{k=0}^{\infty} R_{k}(\lambda, \sigma) \mathcal{P}_{k}(\lambda)$, of the surface measure $\sigma$ on the Kor\'anyi sphere in $\mathbb{H}^n$, in a certain ``high frequency" region, that is, when $2(2k+n) |\lambda| \gg 1$; which seems to be new in the literature. We also show that the positive upper density cannot be qualitatively improved further.